A240509 Least number k > 0 such that n^k - (n-1)^k - ... - 3^k - 2^k is prime, or 0 if no such k exists.

1, 2, 2, 0, 0, 4, 5, 0, 0, 10, 27, 0, 0, 13, 18, 0, 0, 26, 57, 0, 0, 16, 35, 0, 0, 219, 19, 0, 0, 373, 48, 0, 0, 35, 33, 0, 0, 94, 93, 0, 0, 225, 47, 0, 0, 47, 223, 0, 0, 3227, 49, 0, 0, 199, 127, 0, 0, 45, 67, 0, 0, 65, 123, 0, 0, 103

Offset: 2

Author

Derek Orr, Apr 06 2014

Keywords

• nonn
• more

Comments

a(n) = 0 if and only if n == 1 or 2 mod 4. This is because of the parity of the number given. For n = 1, 2, 5, 6, 9, 10, 13, 14,... any k-value will return an even number. Thus, it will never be prime. The only exception is for n = 1, where it will return 1, still not a prime. Further when n = 2, it only returns even numbers; however, 2 is a prime and thus, a(2) = 1.

A prime number is in the sequence A000040.

Next term a(68) is most likely > 5000.

Examples

			7^1 - 6^1 - 5^1 - 4^1 - 3^1 - 2^1 = -13 is not prime. 7^2 - 6^2 - 5^2 - 4^2 - 3^2 - 2^2 = -41 is not prime. 7^3 - 6^3 - 5^3 - 4^3 - 3^3 - 2^3 = -97 is not prime. 7^4 - 6^4 - 5^4 - 4^4 - 3^4 - 2^4 = 127 is prime. Thus, a(7) = 4.
		

Crossrefs

Cf. A042963, A057468, A138699.

Programs

PARI

a(n)=for(k=1,5000,if(ispseudoprime(n^k-sum(i=2,n-1,i^k)),return(k)));
n=1; while(n<100,print1(a(n), ", ");n+=1)